Interferometric lithography system

The researcher on several occasions found out that people are dismayed at the mention of mathematics (geometry) with special needs children. The attitude also is that people wonder as to why a person with visual disability should think of offering mathematics.

2.2.1 Interest of Learners with Visual Impairment in Mathematics

Abraham Nemeth was born blind, and later became a renowned scholar of mathematics. He created the Nemeth code for Braille mathematics in America, and became a Professor of mathematics for 30 years at the University of Detroit. Nemeth retired in 1985 (Nemeth, 2006). He once was told that the field of mathematics was too difficult for people without usable vision. His experience was that after he graduated from high school, he entered Brooklyn College. Nemeth wanted to major in mathematics, but his guidance counsellors insisted that it was too technical a subject for a blind person. His guidance counsellors complained that mathematics notation was too specialized, that volunteer or even paid readers would be difficult to recruit and that he would never get a job in the field of mathematics. According to Nemeth he complied with them and he backed down. After obtaining a degree in psychology and was not satisfied with that, he went on to take a second degree in Mathematics area of his

interest. This was a demonstration of the effect of positive attitude towards developing intrinsic interest in mathematics that enabled a learner with visual impairment to be able to read mathematics despite discouragement by other people.

Ramesh (2006) discovered that children with visual impairment could learn mathematics when they were taught in an appropriate manner by making necessary adaptations in the curriculum without altering the learning outcomes. Ramesh further stated that with special efforts, and conviction of the teacher teaching the children with visual impairment mathematics, an effective application of methodology, and useful adaptation techniques enabled children with visual impairment achieved the same learning outcomes. This was why special education was involved in the teaching of mathematics to special needs children.

However, the learning of mathematics especially geometry is always considered to be difficult and is a complex process even for the non-disabled children. Centre for Teaching/ Learning of Mathematics [CTLM], found out that worldwide, mathematics has the highest failure rates, and lowest average grade achievements. The Centre also added that almost all students regardless of the school type or grade could not perform in mathematics on par with their intellectual abilities (as cited by Ramesh, 2006). Ramesh was of the view that if mathematics for the sighted children would be difficult, the same for children with visual impairment would further be compounded due to loss of vision.

It would even be worse in the case of geometry. However the teaching of mathematics to children with visual impairment has undergone transition over a period of time, resulting in optimistic views toward learning of mathematics by children with visual impairment.

The researchers‘ experiences with teachers in the primary, post primary and tertiary institutions who thought it was impossible for children with visual impairment to offer mathematics showed that they were beginning to see how feasible it was and were

beginning to be optimistic about it. In reality, it is not the difficulty of the child with visual impairment to understand mathematical or geometry concepts, but it is the difficulty of the teacher teaching mathematics or geometry to make suitable adaptations in teaching the concepts (Ramesh, 2006).

O‘Connell and Johnson (2006) stated that in 2001, President Bush of the United States of America signed the No Child Left Behind (NCLB) legislation. This was incorporated in the Individual with Disability Education Act (IDEA). O‘Connell and Johnson further stated that no child was exempted from offering mathematics instead it was made as a compulsory requirement for graduation at different levels in different states in America. They cited example with California that, all students now need mathematics in order to pass the Californian High School exit examination to graduate with a diploma.

Similarly the Nigerian National Policy on Education has promised the provision of adequate education to persons with disabilities (FGN, 2004). There can never be adequate education without mathematics. The need to pass mathematics at the credit level to qualify for admission into the Nigerian Universities needs not be overemphasized. A research into teaching and learning of mathematics by special needs children should be viewed with great importance in the world in general and in Nigeria in particular.

Ramesh (2006) admonished that the effective application of methodology, and useful adaptation techniques would enable children with visual impairment to achieve the same learning outcomes. Special education is an education that meets the needs and provides the knowledge and skills that are necessary in achieving general educational goals for people with impairment (Sardegna & Paul, 1991). Special education is all about adaptation and modification of general or standard curriculum. It involves

changing the educational content, methods of instruction or teaching techniques, instructional materials, environmental factors (O‘Connell & Johnson, 2006)

The Department for International Development [DFID] (2006) defined special educational needs in Africa as the extra support provided by schools, colleges, universities, government education and health department for students who may be unable to follow a mainstream curriculum because of a learning disability. The children that benefit from the special education include: children whose general intellect and ability to learn is significantly restricted compared with that of majority of their peers. It covers a considerable range: from children who can communicate readily in words and who can read and write, to children with no ability to use language. DFID further stated that Special Education also includes the provision of special institutions or schools that cater for a specific sensory impairment (eg blindness). These may include schools for the visually impaired and hearing impaired or unit attached to mainstream schools.

Special education therefore is about children or youth who have impairment of different types and severity, which cause learning difficulties. Children with disabilities have peculiar learning difficulties that except the modified instructional methods and materials are used, none of the regular teachers can easily understand them or teach them mathematics and other subjects especially at the lower level where a lot of concentration is on concept development in the subject area. Many friends and colleagues in mathematics and the sciences find it difficult to understand why a person in special education should do a research in mathematics since they cannot see any relevancy between special education and mathematics

Researches in mathematics in the area of special education are on the increase.

A lot of efforts are being made in the various areas of special education to see that all categories of children with special needs can offer mathematics. The outcome of these

researches was found to be not only useful for children with special needs alone, but have resulted in great improvement in the other children in regular schools (Ramesh, 2006).

2.2.2 Attitude of Learners with Visual Impairment towards Mathematics

Veld (2010) stated that in Netherland, the government encouraged more young people to specialise in mathematics; however the challenge faced was in teaching the subject to persons with visual impairment. It is often said that it is impossible for students with visual impairment to participate in mathematics lessons on equal ground with their sighted peers. Veld observed that students with visual impairment have the same talents as their sighted peers and some have a real talent for mathematics. However veld stated that many teachers and other professionals see mathematics as unsuitable option for learners with visual impairment. They are unaware of the possibilities that even higher level mathematics could be taught to learners with visual impairment. This negative attitude in turn influences the beliefs of the learners and so they admit that mathematics is not a course for them to offer. Often this negative attitude is as a result of ignorance. Some people have wondered as to how a person with visual impairment would tackle mathematics formulae or read and draw diagrams without sight.

Velera (2010) reported that in Peru; Mathematics is a core subject in primary and secondary education. However there was reluctance to accepting children with visual impairment in their classroom for mathematics. Velera in an interaction with many of the teachers documented common questions that came from these teachers who believed that children with visual impairment could not do mathematics and this also in turn made the children to strongly believe that mathematics was not a subject for them to offer. Some of the questions the teachers asked were: How could you introduce mixed numbers and fractions? How would you introduce diagram? How would you introduce

calculations with mixed terms? How would you introduce computation to a blind student? How would you teach graphs to a blind student?

There are certain barriers to teaching and learning of mathematics by learners with visual impairment. Most of these barriers are responsible for the learners forming negative attitude towards mathematics. Chander (1992) stated that although lack of vision is a constraint to learning mathematics, but to a certain extent, inappropriate teaching methods and non-availability of learning materials make the problem. Gupta (1999) thought that fear and negative attitude towards mathematics was due to horrifying myths people held about mathematics. Some of these myths are that mathematics is difficult. Recently a student who had credits in all her subjects in senior secondary school certificate except mathematics rewrote mathematics examination three times but did not pass. When she went to write the examination for the fourth time, as she stepped into the SSCE examination hall to write mathematics again, she collapsed and fainted. She was on the hospital bed for two weeks.

Veld (2010) mentioned some of the factors that cause difficulties for persons with visual impairment in learning mathematics as such they develop negative attitude towards the subject. There are very few people with the skills in teaching learners with visual impairment mathematics. Mathematics Braille code is difficult to master so there is lack of Braille mathematics code knowledge.

Vel in mentioning some of the factors that are responsible for children with visual impairment having negative attitude towards mathematics states that the teachers do not have the time and patience to give the amount of individual explanation children need or the time to check that the student has understood the concept. A frustrating aspect of teaching and learning mathematics is in the lack of Braille mathematics text books. They are more difficult to produce and the codes are difficult to interpret. Vel

reported a case of a student with visual impairment who in a mathematics class could not see the board but was asked to sit in front. She was passively pretending nothing was wrong and not participating because the teacher did not communicate with her. The student believed if when she was sighted and could barely pass mathematics when she could use paper and pencil how on earth could she do mathematics without those necessities? Her anxieties were reinforced by the discouraging comments by some teachers who informed students with visual impairment that they do not belong to their mathematics classes.

All these have helped in making learners with visual impairment develop negative attitude towards mathematics. Hill and Jurmang (1993) created an acronym that mathematics teachers of learners with visual impairment should always keep in their minds ―KISS‖. It simply means ―KEEP IT SUPER SIMPLE‖. It takes a lot of time in presenting the simplest concept in mathematics especially geometry with a learner with visual impairment.

2.2.3 Achievement of Learners with Visual Impairment in Mathematics

In mathematics for the child with visual impairment, he needs the opportunity to learn tactile discrimination, names of shapes, Braille numerals, numeral names and how to make and interpret tactile graphs. The child can explore objects, discuss their properties and learn counting strategies that promote accuracy (Petreshene, 1985).

According to Ramesh (2006) one of the major objectives of teaching mathematics is to develop computation skills, to emphasize logical thinking and to enable the child to participate in day-to-day activities of the family and community. To practically support that, Petreshene (1985) stated that children with blindness can participate in early mathematical experiences through encouraging the child to involve in hands on activities including such chores as table setting that promote counting and

one to one correspondence. All these concepts are present in both orientation and mobility and in mathematics.

Petreshene further stated that a ―number line‖ made of stairs could be used to teach the concept of signed numbers. The teacher should take the students to a landing between floors, with stairs going up and down. The landing is zero. The stairs going up from the landing are positive numbers, while the stairs going down are negative numbers. Have the student go to ―positive seven‖ (+7), or seven steps up. Then ask the student to add a ―negative nine.‖ To do this, ask the student which way is negative?

When the student responds ―down‖, asks the student to move down nine steps, and to tell where the student is in relation to the landing [he is at ―negative two‖ (-2)]. The student can relate what he has experienced with the number sentence ―+7+-9=-2.‖ This process can be continued with additional addends of both positive and negative value. In orientation and mobility, this is the skill of ascending and descending a staircase.

In order for students to calculate using the four basic operations, they must first have developed basic concepts (including more, less, many, etc.), one to one correspondence, the concept of sets, and basic number sense. According to Smith (2006) as students begin to learn to calculate, the following teaching considerations should help:

Emphasize concept development rather than process or rote memorization. To observe that angles are not affected by the length of their rays, students can place items such as the long cane perpendicular to the floor and use their Braille protractor to measure the right angle formed, noting that one ray (the cane) is much shorter than the other (floor).

To demonstrate the concept of correct movement of the decimal point in metric, the student can use a paper plate as a ―dancing decimal point.‖ A group of students can

stand in a row, each with an assigned number. The paper plate decimal point is moved between each student, either to the left or to the right, depending on whether there is a change to a larger metric unit or to a smaller metric unit. For example, students could be named with each of the following digits: 2, 5, and 9. To have the number represent 259 meters, the decimal point plate can be placed to the right of the 9. Then, to change to kilometres, the decimal point plate is moved three places to the left, before the 2, representing 2.59 kilometres, and so on. If there are not enough students, lined up chairs could also be used.

A lot of researches have been done in mathematics for primary school children with visual impairment in the area of mathematical computation. It is very important that students see mathematics, and the calculations they perform, as part of their daily life. Providing opportunities to apply basic concepts and operations in daily activities will reinforce students' skills and motivate them to progress in mathematics.

(Petreshene1985).

Napier (1974) found that the step-by-step progression in content from grade to grade throughout elementary schools in terms of mathematics for both seeing and children with visual impairment is the same. However, the methods and materials are likely to be different. An important goal of elementary school mathematics is the development of number concepts or numeracy (Hart, 1989). Kamii (1981) stated that the numeration systems have been developed to record numbers and to perform calculations.

For example Duncan, Gapps, Dolciani, Quast, Zweng and Gleason (1975) in presenting basic concepts and skills in primary school mathematics show different ways in which numbers are presented. They stated that numbers or sets could be joined like

4 plus 3 equal 7 4 + 3 = 7

The 3 and the 4 are addends. When a child with visual impairment is to do this operation on the abacus he needs to clearly identify the addends. First the child needs to set the first addend and then does the operation of the addition by setting the second addend to the first addend.

Numbers or sets could also be separated.

7 minus 4 equal 3 7 - 4 = 3

―7‖ is the sum.

The child with blindness should understand that in subtraction, he first set the sum or what he first set is the sum and it is from the sum that he deduct the given figure to get the second figure which if added to the given number, it will give exactly the sum he has just first in doing the separation. Similarly, when numerals are presented as follows:

4 + 2 = 6 and 8 - 2 = 6 are equations.

Numbers could be represented as inequalities 5 < 7 or 7 > 5, They could be families of facts (13 + 5 = 18),

Expanded numerals 50 + 6 or compact numeral (56)

Addition, subtraction or any of the four basic mathematics computations done in equation or vertical form order of numbers etc. Kamii (1981) stated that computations in this system are eminently sensible and easy to perform. This aspect of mathematics is important and crucial in the life of the child with visual impairment. The ability to read and write the Braille mathematical notations for the five mathematical notations and the Braille symbols for inequalities are derived from the literal Braille. Braille numeracy or numbers are identical with the first ten letters of the alphabet and are preceded by a special sign in Braille. This means that a child with blindness has to learn two symbols for each number (Tooze, 1973). The implication of this for a learner with visual

impairment is that the same ten letters of the alphabet which make the first ten Arabic numerals are the ones dropped down in position and they make up all the punctuation marks in English Braille and the drop letters are preceded by certain Braille dots to make the basic mathematical symbols. Also these dropped ten letters of the alphabet which when preceded by the numeral sign form the first ten Arabic numbers if dropped and preceded by the numeral sign, will form fractions. When a learner with visual impairment attempts to record arithmetic on the Braille writing frame, it is necessary for him to reverse his paper in order to read what he had written. This makes the recording of figures and computation of mathematics difficult or almost impossible for a learner with visual impairment. Also mathematical computation is difficult for a learner with visual impairment to be done vertically.

For the past several hundred years mathematics has been typically presented on the printed page in two dimensional format with symbols printed not only horizontally left or right across the page but also vertically above and below in superscripts, and subscripts. Admittedly this allows the traditional printed form of mathematical expressions to be quite compact. Tooze (1973) stated that this makes it extremely difficult to record mathematics in Braille. He stated that they felt that their area of inquiry should be into more efficient ways of recording mathematics Braille (p.63).

Sighted children use Hindu - Arabic numbers or figures. According to Hart (1980) the Hindu - Arabic numeration system enables children to deal with types of mathematics, which were utterly impossible. For this when you compare using Roman numeral with the Arabic numeral in terms of difference is like between walking and flying. Hart (1980) stated that by analogy mathematics is fundamentally the same whether written in the German, French, Arabic, Chinese, Spanish, Greek, or English (p.1).

Similarly Kamii (1981) stated that when you compare computation in the past medieval times the modern algorithms make the arithmetic operations (addition, subtraction, multiplication and division) accessible to almost everyone. It is therefore much more convenient computing using the Arabic numeral at the primary level.

However, the graphic mark or squiggles in Arabic numbers for sighted are not the same with the shape of embossment in Braille mathematics notation. This creates some differences for a child with visual impairment in recording numerals and in mathematics computation. This situation has set experts in special education for ages in search of the equivalence of paper and pen or pencil for a figure and conveniently computing figures in mathematics.

Common Basic concepts that are found both in mathematics and orientation and mobility skills: (and language to describe them) Size and space relations ―bigger than…‖ ―Less than…‖ Sequencing (ordering) by size, by quantity 1 to 1 correspondence in counting, classification – matching to sorting based on certain characteristics. Others are early awareness of shape and size and awareness of surfaces and faces - pre-area, sorting of three-dimensional shapes and sorting of two-dimensional shapes. Others are capacities sets grouping. seriation, length, pattern – arrangements, coordinate – peg, area, time etc.

Salisbury (1974) warned that emphasis should be laid at the primary level on understanding the properties of the basic operations in mathematics. This comes after basic concepts are developed. Salisbury further stated that the importance of a good foundation in the fundamentals of mathematics cannot be too greatly stressed in the education of learners with visual impairment (p.20).

A simple observation of special schools for children with visual impairment shows that, although there are teaching aids that could be used for practical presentation

of mathematical concepts they are not used. Children with visual impairment lack spatial concept due to lack of sight. This calls for predisposing the child with visual impairment to basic mathematics concept before exposing the child to formal mathematical concepts. The basic addition - subtraction facts are those with both addends less than 10 (Duncan, Gapps, Dolciani, Quast, zweng, & Gleason 1973). When one uses these addition and subtraction facts and the place value system of numeration, one can add or subtract numbers of any size. All that is needed there is to memorize them. For example:

2 + 1 = 3 2 + 2 = 4 4 + 3 = 7 7 + 2 = 9 3 - 1 = 2 7 - 4 = 3 etc.

This principle is used when using the abacus for mathematical computation by the learner with visual impairment. In the use of abacus sometimes you add more than what you wanted to add so you must know how much more you have added in excess and then reduce the excess. Similarly you could subtract more than what you had desired to subtract so as to balance your subtraction. Where you have subtracted in excess, you add back the excess number you subtracted. Duncan, Gapps, Dolciani, Quast, zweng and Gleason presented the following basic concepts in mathematics:

Two place addition. Use two steps to add. (a) e.g. 47 and 26 7 + 6 = 13. Write 3 in the one‘s place, I in the tens place.

47 + 26 73

1 + 4 + 2 = 7

Write 7 in the tens place.

47 + 26 73

Different method 47 + 26 13 60 73

In using the Cranmar abacus for mathematical computation, the different method is used.

Three place addition 5 2 4

1 9 8 1 2 1 1 0 6 0 0 7 2 2

ii. Properties of addition

(a) The commutative property.

The order of the addends does not affect the sum.

4 + 3 = 7 3 + 4 = 7

These are concepts that learners with visual impairment need to be told because they do not have easy access to these computations as to enable them easily see the patterns involved.